When \((p\log ^2n)/n\rightarrow 0\) , \(\mathbb {E}\Vert  \Sigma -\mathbb {E}_{\Sigma |\mathbf {X}}\Sigma \Vert ^4=O(((p\log n)/n)^2)=o(p/n)\)  based on equation (A.5) in Lemma A.6 of [1]}. As will be shown later, the dominant term is in \(\Theta (p\Vert \theta _0\Vert ^2/n)\) , therefore \(\mathbb {E}\Vert  \Sigma -\mathbb {E}_{\Sigma |\mathbf {X}}\Sigma \Vert ^4\Vert \theta _0\Vert ^2\)  is only a remainder term. Furthermore,
\(&&\mathbb {E}_{\Sigma |\mathbf {X}}\Vert (\mathbb {E}_{\Sigma |\mathbf {X}}(\Sigma +\lambda \mathbf {I}_p )^{-1}\Sigma \theta _0)-(\Sigma +\lambda \mathbf {I}_p )^{-1}\Sigma \theta _0)\Vert ^2\\&=&\lambda ^2\mathbb {E}_{\Sigma \mathbf {|}\mathbf {X}}\Vert (\mathbb {E}_{\Sigma |\mathbf {X}}(\Sigma +\lambda \mathbf {I}_p )^{-1}\theta _0)-(\Sigma +\lambda \mathbf {I}_p )^{-1}\theta _0\Vert ^2\\&:=&\lambda ^2\psi (\lambda ).\) 
